#454433

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3062	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(9,10,16,22)(11,23,28,35)(12,18,31,38)(13,32,30,19)(14,21,15,20)(17,27,36,26)(24,37,25,40)(29,33,39,34);(5,6)(7,8)(9,11,15,27,16,28,14,26)(10,17,21,35,22,36,20,23)(12,29,13,25,31,39,30,24)(18,37,19,34,38,40,32,33);(2,4)(6,8)(9,12)(10,18)(11,24)(13,14)(15,30)(16,31)(17,33)(19,20)(21,32)(22,38)(23,37)(25,28)(26,29)(27,39)(34,36)(35,40);(9,13,15,31,16,30,14,12)(10,19,21,38,22,32,20,18)(11,25,27,39,28,24,26,29)(17,34,35,40,36,33,23,37)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361629);